A Fractal Function Related to the John–Nirenberg Inequality for Q α(Rn)

Yue, Hong

Canadian journal of mathematics, Tome 62 (2010) no. 5, pp. 1182-1200

Résumé

A borderline case function $f$ for ${{Q}_{\alpha }}\left( {{\mathbb{R}}^{n}} \right)$ spaces is defined as a Haar wavelet decomposition, with the coefficients depending on a fixed parameter $\beta \,>\,0$ . On its support ${{I}_{0}}\,=\,{{\left[ 0,\,1 \right]}^{n}},\,f\left( x \right)$ can be expressed by the binary expansions of the coordinates of $x$ . In particular, $f\,=\,{{f}_{\beta }}\,\in \,{{Q}_{\alpha }}\left( {{\mathbb{R}}^{n}} \right)$ if and only if $\alpha \,<\,\beta \,<\frac{n}{2}$ , while for $\beta \,=\,\alpha $ , it was shown by Yue and Dafni that $f$ satisfies a John–Nirenberg inequality for ${{Q}_{\alpha }}\left( {{\mathbb{R}}^{n}} \right)$ . When $\beta \,\ne \,1$ , $f$ is a self-affine function. It is continuous almost everywhere and discontinuous at all dyadic points inside ${{I}_{0}}$ . In addition, it is not monotone along any coordinate direction in any small cube. When the parameter $\beta \,\in \,\left( 0,\,1 \right)$ , $f$ is onto from ${{I}_{0}}$ to $\left[ -\frac{1}{1-{{2}^{-\beta }}},\,\frac{1}{1-{{2}^{-\beta }}} \right]$ , and the graph of $f$ has a non-integer fractal dimension $n\,+\,1\,-\beta$ .

Détail

DOI : 10.4153/CJM-2010-055-x

Mots-clés : 42B35, 42C10, 30D50, 28A80, Haar wavelets, Q spaces, John–Nirenberg inequality, Greedy expansion, self-affine, fractal, Box dimension

Bibliographie

[1] [1] Chang, D-C. and Sadosky, C., Functions of bounded mean oscillation. Taiwanese J. Math. 10(2006), no. 3, 573–601. Google Scholar

[2] [2] Dajani, K. and Kraaikamp, C., Ergodic theory of numbers. Carus Mathematical Monographs, 29, Mathematical Association of America, Washington, DC, 2002. Google Scholar

[3] [3] Dafni, G. and Xiao, J., Some new tent spaces and duality theorem for Carleson measures and Qα(Rn). J. Funct. Anal. 208(2004), 377–422. doi:10.1016/S0022-1236(03)00181-2 Google Scholar

[4] [4] Dafni, G. and Xiao, J., The dyadic structure and atomic decomposition of Q spaces in several real variables. Tohoku Math. J. 57(2005), no. 1, 119–145. doi:10.2748/tmj/1113234836 Google Scholar

[5] [5] Erdös, P., Horváth, M., and Joó, I., On the uniqueness of the expansions 1 = Pq−ni . Acta Math. Hung. 58(1991), no. 3–4, 333–342. doi:10.1007/BF01903963 Google Scholar

[6] [6] Erdös, P. and Joó, I., On the expansion 1 = Pq−ni . Period. Math. Hungar. 23(1991), no. 1, 27–30. doi:10.1007/BF02260391 Google Scholar

[7] [7] Erdös, P., Joó, I., and Komornik, V., Characterization of the unique expansions 1 = Pq−ni and related problems. Bull. Soc. Math. France 118(1990), no. 3, 377–390. Google Scholar

[8] [8] Essén, M., Janson, S., Peng, L., and Xiao, J., Q spaces of several real variables. Indiana Univ. Math. J. 49(2000), no. 2, 575–615. Google Scholar

[9] [9] Falconer, K., Fractal geometry: Mathematical foundations and applications. John Wiley &amp; Sons, Ltd., Chichester, 1990. Google Scholar

[10] [10] Fefferman, C., Characterization of bounded mean oscillation. Bull. Amer. Math. Soc. 77(1971), 587–588. doi:10.1090/S0002-9904-1971-12763-5 Google Scholar

[11] [11] Fefferman, C. and Stein, E. M., Hp spaces of several variables. Acta Math. 129(1972), no. 3–4, 137–193. doi:10.1007/BF02392215 Google Scholar

[12] [12] Hueter, I. and Lalley, S. P., Falconer's formula for the Hausdorff dimension of a self-affine set in R2. Ergodic Theory Dynam. Systems 15(1995), no. 1, 77–97. doi:10.1017/S0143385700008257 Google Scholar

[13] [13] John, F. and Nirenberg, L., On functions of bounded mean oscillation. Comm. Pure Appl. Math. 14(1961), 415–426. doi:10.1002/cpa.3160140317 Google Scholar

[14] [14] Neri, U., Some properties of functions with bounded mean oscillation. Studia Math. 61(1977), no. 1, 63–75. Google Scholar

[15] [15] Stein, E. M., Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series, 43, Monographs in Harmonic Alalysis, III., Princeton University Press, Princeton, 1993. Google Scholar

[16] [16] Xiao, J., Homothetic variant of fractional Sobolev space with application to Navier-Stokes system. Dyn. Partial Differ. Equ. 4(2007), no. 3, 227–245. Google Scholar

[17] [17] Yue, H. and Dafni, G., A John–Nirenberg inequality for Qα(Rn) spaces. J. Math. Anal. Appl. 351(2009), no. 1, 428–439. doi:10.1016/j.jmaa.2008.10.020 Google Scholar